Unitary representations of solvable Lie groups

A NNALES SCIENTIFIQUES DE L ’É.N.S.

L. P ^UKANSZKY

Unitary representations of solvable Lie groups

Annales scientifiques de l’É.N.S. 4^e série, tome 4, n^o4 (1971), p. 457-608

<http://www.numdam.org/item?id=ASENS_1971_4_4_4_457_0>

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1971, tous droits réservés.

L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.

elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

4® s6rie, t. 4, 1 9 7 1 , p. 4^7 ^ 608.

UNITARY REPRESENTATIONS OF SOLVABLE LIE GROUPS (

)

By L. PUKANSZKY.

Du kennst doch das Schillersche Gedicht " Spruch des Konfucius " und weisst, dass ich da besonders die Zeilen liebe : Nur die Fulle fuhrt zur Klarheit und im Abgrund wohnt die Wahrheit.

N. BOHR, quoted in W. HEISENBEBG, Der Teil und das Game (R. Piper Co., Munchen, 1969, p. 284).

TABLE OF CONTENT.

Pages.

CHAPTER I : The transitive t h e o r y . . . 464

S u m m a r y . . . 464

1. Some factor representations of central extensions by a 1-torus of free abelian g r o u p s . . . 465

2. Some factor representations of central extensions by a 1-torus of direct products of free abelian groups and vector g r o u p s . . . 474

3. The representations of the transitive t h e o r y . . . 480

4. Preliminaries on holomorphic i n d u c t i o n . . . 489

5. Computation of the Mackey g r o u p . . . 497

6. Orbits and r e p r e s e n t a t i o n s . . . 500

7. Description through holomorphic induction. Theorem 1 . . . 505

CHAPTER II : Generalized orbits of the coadjoint r e p r e s e n t a t i o n . . . 512

S u m m a r y . . . 512

1. Orbits of linear solvable algebraic g r o u p s . . . 513

2. Regularization of the orbits of the coadjoint representation. . . . 521

3. Construction of the torus bundles. I. Algebraic p r e l i m i n a r i e s . . . 523

4. Orbits of the coadjoint representation of a nilpotent g r o u p . . . 525 (l) The present paper was partially supported by a grant from the National Science Foundation.

Ann. ^c. Norm., (4), IV. — FASC. 4. 58

Pages.

5. Construction of the torus bundles. II. Topological p r o p e r t i e s . . . 529

6. Definition and normal form of the o b s t r u c t i o n . . . 532

7. The generalized o r b i t s . . . 539

8. Exemple of a nontrivial o b s t r u c t i o n . . . 541

CHAPTER III : The nontransitiue t h e o r y . . . 544

S u m m a r y . . . 544

1. Invariant measures on generalized o r b i t s . . . 545

2. Construction of the central factor representations. Theorem 2 . . . 548

3. Central decomposition. Theorem 3 . . . 552

CHAPTER IV : Structure of the regular r e p r e s e n t a t i o n . . . 566

S u m m a r y . . . 566

1. On the irreducible unitary representations of simply connected nilpotent Lie g r o u p s . . . 568

2. The non simply connected c a s e . . . 569

3. A c o u n t e r e x a m p l e . . . 574

4. Construction of some rational s e m i - i n v a r i a n t s . . . 575

5. Preliminaries on central distributions on nilpotent g r o u p s . . . 580

6. Fourier transforms of central distributions.. . . . 582

7. Construction of semicanonical trace. Theorem 4 . . . 586

8. Size of the collection of the type I or type II o r b i t s . . . 594

9. Triviality of the type I or type II component. Theorem 5 . . . 602

BIBLIOGRAPHY . . . 606

SOME NOTATIONAL CONVENTIONS. . . . 607

INTRODUCTION (²).

The investigations of the present paper started with an examination by the present author, through special examples, of the possibility to extend the recent theory of type I solvable Lie groups by L. Auslander and B. Kostant (c/*. [1]) to arbitrary Lie groups. These authors, carrying forward by an essential step the line of research started by A. A. Kiril- lov [22] and continued by P. Bernat [3], using results by G. C. Moore, gave a neccessary and sufficient condition in order that a connected and simply connected solvable Lie group be of type I. Furthermore they provided in this case a complete description, by aid of the orbit space of the coadjoint representation, of the dual. Thus, in particular, if G is such a group and fl is its Lie algebra, then G is of type I if and only if : (1) Any orbit of G on fl' (== dual of the underlying space of g) is locally

(•2) The Introduction and the Summary, in front of each chapter, intends to give only an outline of the results of this paper. For a precise formulation of these as well as for complete references to the literature we refer to the corresponding point of the detailed discussion.

closed and (2) The de Rham class of the canonical 2-form is always integral (and hence zero; cf. Th. V. 3.2, loc. cit.). Two examples, due to J. Dixmier {cf. [10]) and F. I. Mautner resp., of solvable groups which are not of type I, are particularly well known in the literature (for the definition of these cf. the Summary of Chapters I and II). From among these Dixmier's group satisfies the first of the above conditions but not the second, Mautner's violates the first, but satisfies the second. A closer inspection of Dixmier's example led us to the conclusion, that by aid of a natural extension of the procedure of Auslander and Kostant one can associate to each orbit in the general position a well determined factor representation of type 11^. More importantly it turned out, that this relationship admits a description modelled after Kirillov's theory of characters of a connected and simply connected nilpotent group. Let G be such a group and g its Lie algebra. We can identify the underlying manifold of G to that of % by means of the exponential map. The measure, corresponding on G to a translation invariant measure dl on g is biinva- riant. Let T be an irreducible unitary representation of G, y an element of C^ (fl) and let us form the operator T (y) = fcp (;) T (?) dl. It is of

^Q

trace class, and Kirillov's formula, which is the natural analogue {cf. [30], p. 258-264) of the character formula of H. Weyl for compact semi-simple groups, provides the following expression for its trace. Let us write

<^,r)> (^€fl, ^'Gfl') for the canonical bilinear form of the underlying abelian group of g. We define the Fourier transform of <p by

^(o-r^xu^ (^ ^/ ).

^Q

Then there is a uniquely determined orbit 0 of G on g' such that (1) T^TCp))^^)^

^o

where dv is an appropriately normalized invariant measure on 0. Let us observe, incidentally, that in the case envisaged here, the proof of the abso- lute convergence of the right hand side is relatively simple. Conversely, to each orbit 0 there is a unitary equivalence class, corresponding to 0 by virtue of formula (1). In other words (1) can be used to define a bisection between elements of the orbit space and of the dual of G resp.

{cf. for all these e. g. [29]). Returning to the example of Dixmier we found, that with the factor representations we constructed (1) substan-

tially retains its validity, provided on the left hand side by Tr (T (y)) we mean the value on T (y) of an appropiately normalized trace in the sense of the 11^ factor generated by T. We obtained similar conclusion for the group of Mautner with the difference, that in place of 0 we had to substitute closures of orbits of the coadjoint representation of G.

The main consequence of the above observation for us was a concrete suggestion, that perhaps for all connected solvable Lie groups the left (or right) regular representation is a continuous direct sum of semi- finite factor representations, or what amounts to the same, the left (or right) ring, that is the von Neumann algebra generated by the left (or right) regular representation is semifinite. Let us recall, that this was shown by I. E. Segal to be the case for any separable locally compact unimodular group (c/*. [34]) but was disproved by R. Godement in the general case. This conclusion, in fact, imposes itself by assuming, that for any connected solvable group, too, sufficiently many semifinite factor representations can be constructed, such that the essential features of (1) be preserved, and by observing the mechanism of the Plancherel formula in the nilpotent case. In fact, let us write A = g'/G, and let us set T>., 0\ and dv\ resp. for the objects, as in (1), corresponding to X e A . Then to show, that the representations { T ^ A e A } provide a central conti- nuous direct sum decomposition of the left regular representation, one has to prove, that the value 9 (0) of y at zero can be reconstructed from the values Tr (T), (y)) by aid of a formula of the type

(2) cp(0)=fTr(T,(cp))d^).

^ \_A

But if dV is an appropriately normalized translation invariant measure on g', we have

^=^(l')dl\

From this we conclude, that to obtain a formula as (2), it suffices to repre- sent dV as a continuous direct sum of the measures dv\ by aid of a measure dy. on A.

Although much progress has recently been made toward a clarification of the possibilities of a formula as (1) for type I groups (cf. [15]), unfor- tunately already in this case any attempt to obtain a theory as for the nilpotent groups is confronted with great difficulties. Their reasons, among others are, that a bijection along Kirillov's lines is limited to

groups with simply connected orbits, and that it seems to be exceedingly difficult to establish the convergence of integrals as in (1) for a suffi- ciently ample family of functions. We wish to observe, incidentally, that these problems do not at all appear to increase by abandoning the assumption, that our group be of type I. In this fashion, to follow up the indications carried out above, we had to look for a different tool which we found in the theory of quasi unitary algebras due to J. Dixmier (c/- [7])- ^s a ^^l^ we succeeded in establishing the purely global result, that the left (or right) ring of any connected but not necessarily simply connected Lie group carries a faithful trace (3), such that the corresponding family of generalized Hilbert-Schmidt operators generates the whole ring (c/*. Theorem 4, Chapter IV of this paper). Hence, in particular, the left (or right) ring of any group of the said sort is semi- finite. This conclusion has been shown in the mean time by J. Dixmier to retain its force for an arbitrary connected topological group (c/*. [14]).

This result of ours, however, leaves open the problem of the possibility of an « orbitwise » theory of factor, representations. One can namely raise the question, if the procedure of Auslander and Kostant, through an appropriate modification, leads to a class of factor representations, which can claim some special interest. In this paper we show, that this is indeed the case as already suggested, incidentally, by the examples of Dixmier and Mautner discussed above. Our main result concerning this point {cf. Theorems 2 and 3, Chapter III) provides a family of factor representations parametrized by certain geometrical objects, genera- lizing the orbits of the coad joint representation in such a fashion, that the regular representation admits a central continuous direct sum decom- position involving only these representations. The necessity to consider more than one representation for one orbit, and thus to go beyond these in a search for objects parametrizing the dual, arises already in the case of the universal covering group of the motion group of the Euclidean plane. For the general type I group, according to the algorithm of Auslander and Kostant, the irreducible representations, belonging to the same orbit, can be parametrized by a torus, the dimension of which is equal to the first Betti number this orbit. Our construction proceeds in two major steps. First (c/*. Chapter I) we associate with any orbit a family of semifinite factor representations, the members of which are in a one to one correspondence with the underlying set of a torus. The dimension of the latter, however, is in general different from that of the

(3) For our definition of the trace c/. e. g. Section 7 in Chapter IV.

type I theory. Example for this situation is given by an orbit in the general position of Dixmier's group. Here our torus is zerodimensional, while the Betti number in question is 2. If all orbits are locally closed, as is the case, in particular, for the type I groups, the collection of all these representations already provides a central decomposition of the regular representation. For a type I group this step essentially repro- duces the algorithm, defining the orbit-representation relation, of Auslander and Kostant. The only difference is, that our representations are not necessarily irreducible ones, but one or 4- oo fold multiples of such repre- sentations. If, however, there are orbits, which are not locally closed, as in the case of the group of Mautner, to obtain the « central components » of the regular representation a more involved construction is necessary.

In Chapter II we introduce a generalization of the orbit concept leading to certain sol vmanif olds, which are transformation spaces of our group, such that any orbit of the latter is dense. Also, these spaces carry inva- riant Borel measures. From here, by virtue of a classical principle (c/*.

Lemma 2.3.1, Chapter III) we obtain our « central factors »in Chapter III by forming continuous direct sums over the said manifolds of appropriate subcollections of the representations of Chapter I. Groups, violating simultaneously both conditions of Auslander and Kostant, may at this point display additional difficulties, not indicated by the examples of Dixmier and Mautner {cf. Section 8, Chapter II). Finally, using the previous theory we show, that if our group is simply connected, the left (or right) ring coincides with its type I or type II component {cf. Theorem 5, Chap- ter IV). In other words, the left (or right) regular representation of any such group admits a central-continuous direct sum decomposition into factor representations, all of which are either of type I or of type II only.

Let us also observe, that our results imply the necessity of the conditions of Auslander and Kostant quoted at the start. In fact, at once one of these is not satisfied, there appears in our list a factor representation, which is not of type I.

It is clear from the beginning, that our construction cannot aim at a complete classification of the factor representations of these groups.

For example, in the case of the groups of Dixmier and Mautner our proce- dure provides semifinite factor representations only. On the other hand, since these groups are not of type I, by virtue of the results of J. Glimm they admit type III representations. We shall say, that the unitary representation T is of trace class, if on the von Neumann algebra R (T) it generates there exist a faithful, normal and semifinite tra.ce (3), such that the set of generalized Hilbert-Schmidt operators in the range of the associated representation of the group C* algebra generates R (T)

{cf. Section 7, Chapter IV). For instance, by what we saw above, the left (or right) regular representation of a connected solvable Lie group always has this property. Our results imply, that an « overwhelming » majority of the representations appearing in our list are trace class representations and hence, in particular, generate semifinite factors. But we leave in this paper the problem of an individual characterization of these repre- sentation open. While admitting, that certain points of the following programm, at the present stage of the research, might appear overly ambitious, we still believe, that ultimately it turn out, that our represen- tations, up to quasi-equivalence, give precisely the collection of all trace class representations. In addition we conjecture, that the factors they generate are always approximately finite. The significance of the last point is, that in this fashion one could show, that by considering factor representations, which are not of type I, one does not get involved in the algebraic type problem of factors. Or, to put it more succintly, this widening of the view point should not place one in a situation worse, than in the type I theory. The author is indebted to C. C. Moore for the following suggestion of a collective characterization of our represen- tations. One could try to show, that upon forming the kernels of the associated representations of the group C* algebra, one obtains precisely once each primitive ideal of the latter. Let us observe, that recently R. Howe obtained results along these lines for a class of discrete nilpotent groups {cf. [21]).

As far as the prerequisites for the reading of the present paper are concerned, our exposition of the necessary results of the geometry of orbits of linear solvable groups is self contained. On the other hand, we assume a relatively advanced knowledge of the theory of induced representations by G. W. Mackey. In fact, we shall use the basic results of [23] and [25] often without special reference. For a summary we refer to [2], Sections 9-10 (p. 50-63). Also, some preliminary familiarity with the notion of holomorphic induction is necessary {cf. the references of Section 4, Chapter I). The reader is advised to consult carefully the list of notational conventions at the end of the paper.

The results of Chapter IV, Sections 1-7 were announced in [33], those of the rest of this paper in the author's conference at the International Conference of Mathematicians, Nice, September 1970.

The author is much indebted to B. Kostant for introduction in his joint work with L. Auslander, and also for discussion in his seminair at the Massachusets Institute of Technology, Fall 1968, of several parts of Chapter IV,

CHAPTER I.

THE TRANSITIVE T H E O R Y .

SUMMARY. — Let G be a connected and simply connected solvable Lie group with the Lie algebra $. As already stated in the Introduction, in this chapter we assign to each orbit of the coadjoint representation of G on g' a family of semifinite factor representations. Our discussion follows at many points the treatment of the type I case by Auslander and Kostant in [1]. One of the major differences appears, however, already at the start. The purpose of Sections 1-2 is to discuss certain factor represen- tations of a group, which is central extension by a one dimensional torus T of a direct product of a free abelian group of finite rank with a vector group. The necessity to consider such groups arises in the following fashion. We denote by L the first derived group of G (or L = [G, G]); L is nilpotent and thus of type I. Let ^ be an irreducible unitary representation of L; then the corresponding Mackey group M^ (c/. the begin of Section 3 for the definition) has the indicated structure. Let r be a group of this class, U the centralizer of the connected center and U^ the center of U. The main result of this part (cf. Proposition 2.1) states, that if 7, is a character of U', such that its restriction to T c U ' is not trivial, then the unitary representation, induced by ^ in r, is a semifinite factor representation, and gives a necessary and sufficient condition that it be of type I.

Let TC be as above, Gn its stabilizer in G, ^e an appropriately chosen projective extension of ^ to Gn, and G^ the corresponding central extension of G by a one dimensional torus.

The collection of the factor representations of this chapter coincides with the family of all representations of the form ind (V®^), where V is a representation, arising by

G^G

lifting to G$ a representation of M^ (== r) obtained as above by aid of a ^c, which on T c U ^ coincides with the conjugate of the identity map of T onto itself, for all possible choice of T: in the dual L of L and /^. In order, that G be of type I, in particular, M^A has to be of type I for all ^ e L. In this case our procedure yields one or infinite foldA multiples of the collection of all irreductible representations of G. Section 3 gives a description, not directly involving the Mackey group, of our representations. It is shown (c/. Lemma 3.5) that each ^ e L uniquely determines a closed subgroup K-n^L, suchA that ^ admits a proper extension p to K-rc, and that our representations coincide with the collection of all representations of the form ind p (for all possible choice of ^ e L and of

A ^ ^G

p e KTT, p | L = ^ ) . We give a necessary and sufficient condition that such a represen- tation be of type I, and that two of them be quasi-equivalent (in which case they are also unitarily equivalent; cf. for all this Lemmas 3.7, 3.8 and Remark 3.3). These considerations do not at all depend on the assumption, that G be solvable, provided L is appropriately chosen. In an effort to bring this to expression, in this section (but only here) we allow G to be an arbitrary simply connected Lie group and take in place of [G, G] a closed, connected, invariant and type I subgroup L, such that G/L abelian.

In Section 4, beside summarizing the necessary prerequisites of holomorphic induction and of the Kirillov theory (this we take for granted), we present the definition of the reduced stabilizer. Let g be an element of g', Gg the stabilizer of g in G with respect to the coadjoint representations, (Gg)o the connected component of the identity and 8^ eg the Lie algebra of the latter. Since G is solvable and simply connected, (G^)o,

too, is simply connected, whence we conclude, that there is a well determined character /^

on (G^)o such that d^g (I) === i (I, g) (/e<^). Let us put G^ = ker (/.^ | (Gg)o); this is an invariant subgroup of G^, and the reduced stabilizer G^ of g is the complete inverse image, in Gff, of the center of G g l G g . Section 5 reproduces the proof of an important result of Auslander and Kostant establishing a relation between the obstruction cocycle belon- ging to T: e L and the Kirillov orbit of ^ in the dual ti' of the underlying space of the LieA algebra t« = [9, 9] of L. Using this in Section 6 we show, that if ^ e L corresponds toA the Kirillov orbit L/ct*' (/et^), and g is any element of 9' such that g \ ti = /, then we have Kn = LGr^» an(^ ln(^ P (p€ ^71, p [ L = ^) is of type I if and only if the group

K^ ^. G

Ggf Gg is finite. This condition can be shown (but we do not carry out this point) to be equivalent to the rationality of the de Rham class of the canonical 2-from on Gg (for a definition of the latter cf. e. g. [30], p. 256). The integrality of this form means, that G^ = Gg and conversely; in order, that G be of type I in particular, this must be valid for all ge.^. The results of this section are used in an essential fashion, among others, in Chapter IV to estimate the «size » of the totality of type I representations in the central decomposition of the regular representation (cf. in particular Proposition 8.1, Chapter IV).

Finally Section 7 brings the construction, along the lines laid down by Auslander and Kostant, of our representations as (in general) holomorphically induced representations.

A .

For a ge.^ let us denote by Gg the collection of all characters of Gg restricting on (Gy)o to /^ (cf. above). Let us put (^ ===^€9 ^Ar^ ^ ls a transformation space of G. One of our main conclusions is, that there is a bijection between the set of all unitary equi- valence classes of our representations and points of ^l/G. Orbits, lying over Gg (g fix in ^/) , in CH are parametrized by points of G^. The underlying set of the latter admits a naturalA identification with the dual of G^/(Gg.)o, which is a free abelian group of finite rank. In the case of a type I group, since here Gg = Gg, Gg/(Gg)o is just the fundamental group of Gg. But, for instance, in the case of the group of Dixmier quoted in the Introduction, the situation is already completely different. This group belongs to the Lie algebra, spanned over the reals by the elements { e/;,l ^ /' ^ 7 } with the nonvanishing brackets

[ei, ^2] = ^7, [ei, 63] == ^4, [ei, 64] == — e.3, [^2, es] = Ce, [^2, Cc,] = — Co.

For a general ge^ we have Gg = (Gg)o, while the rank of Ggl(Gg)o is two. Thus the dimension of the torus, parametrizing the representations belonging to the same G orbit in ig7, is in general different from the first Betti number of the latter.

1. PROPOSITION 1 . 1 . — Let Z be a free abelian group of finite ranky and let us consider a central extension Z of Z by a one dimensional torus T.

Let y be a character of the center Z^ of Z, which, when restricted to T, reduces to the identity map of the circle group onto itself. The unitary represen- tation

ind ^= U

z^z

of Z is a factor representation of finite class which, on T, equals to a multiple of /. U is of type I if and only if the index of 7^ in Z is finite.

Ann. EC. Norm., (4), IV. — FASC. 4. 59

Proof. — a. We recall first {cf. [2],. p. 188), that there exist a skew- symmetric bilinear form a from Z x Z into T (identified with the group of all complex numbers of absolute value one), such that Z is isomorphic to the group of all pairs {z, u) (^€Z), u € T ) with the law of multiplication

(z, u) (z', u') == (z + z\ u.u'.a (z, z')).

Given a subgroup F of Z, we shall write T for the subgroup { ( y , u); y e F, u € T}

of Z. Let us form now the subgroup Zo == { x\ x^Z, (a (a;, y))2 = for all y e Z } of Z; one verifies easily, that Zo coincides with the center Z^ of Z.

We denote by Zi the subgroup of all elements { x\ a [x, y) = 1 for all y in Z }, and by %i the restriction of % to Zi. Finally, we write ^ for the set of all characters of Zo, which on Zi restrict to %i.

LEMMA 1.1. — Putting

V = md^i and Ucp = ind cp (? e ^)

Z i + Z Z o + Z

we have

v = ^ © u ,

t ? € ^

aM6? Uq,, W/I^M restricted to Zo, equals to a multiple of <p.

Proo/*. — Let us observe first, that Zo/Zi is finite; in fact, it is isomor- phic to Zo/Zi, which is of a finite rank and any element in it has the order 2.

We have thus

ind ^i = V © 9

Z^Zo ~

©e^

whence, through induction by stages we conclude, that V = ind %i = ind / ind %i\ = Y © ind cp = V © U..

Z,+Z Z o + Z \ Z ^ Z o / '"'" Z o - h z ~

y e ^ cpe^

Finally, since Zo is the center of Z, Uy on Zo restricts to a multiple of y.

Q. E. D.

b. Let (SL be a countable abelian group and ^ a skew symmetric bili- near form, with values in T (== circle group), on (StxcX. Similarly as

above, we write (Sl for the group defined on the set of all pairs { (a, u); aGdl, u € T } by the law of multiplication

(a, u) (b,u) == (a + b, u.y.(3 (a, b)).

We denote by )^o the character of TCOL defined by %o ((0, u)) = u.

LEMMA 1.2. — With the above notations^ the unitary representation W == ind / o can be described as follows. There is a unitary map from the

T+d

representation space H (W) of W onto L2 (d) (0L being taken with the discrete topology), such that the van Neumann algebra R (W) generated by W goes over into the set of all bounded operators on L2 ((ft) which, with respect to the natural basis can be expressed in matrix form as

{ay-^ P (x, y); x, ye a } (areC, xe a).

The commutant of R (W) goes over into the set of all bounded operators^

which can be written as

{ b y - ^ ^ ( y , x ) ; x,yea\.

Proof. — In the following we shall write a and u in place of (a, 1) and (0, u) resp. (a€^l, u € T ) whenever convenient.

1° Choosing an invariant measure on (9L, by virtue of our definition of 0L, there is a natural isomorphism between the Hilbert spaces L2 (<St) and L2 (T) (g) L2 (0L). Let L and R be the left and right regular represen- tation resp. of 0i on L2 (<9c). Since (0, u)"1 (a, v) == (a, u . y ) , writing R for the ring, generated by the regular representation of T on L2 (T), we see at once, that R (L T) == R(g) I c ( R (L))^, from which, taking into account that R' = R, we conclude, that any operator in the left ring R (L) of <9C can be expressed as a matrix

(1) ( A ^ y ; r K , ? / e A j

the entries taking their values in R. Similar observation applies to the right ring.

2° For w € T and yeL^r) let us write L^ f (u) = f(wu}. Since {z, 1) (^, u) == (z + x, u p (z, a;)), we can conclude, that for f e L2 (<^t),

(L (2) f) (x, u) ^ (L^,.)). f(a: + z, u) [L (2) = L ((z, 1)-%

In this fashion the right ring of 0L coincides with the set of all operators in (1), which commute with any member of the family of operators { L (z); zed }. Similarly, putting for z e d and f^L2 (d) = L^T^L^a)

(R (z) f) (x, u) === (Lp (.,,.)), f (x + z, u)

the left ring is the collection of all elements in (1) commuting with every operator in { R [ z ) ; zGOi }.

3° Let now A be an arbitrary element of (1), and let us write out the condition, that it belong to the right ring. According to what we have just seen, in order that this happen, we must have for all z € Cl and /*€ L2 (dl)

(L (z) A f) (x, u) := (AL (z) f) (x, u).

But

(L (z) Af) (x, u) ^, (LP(,,) A,^,s). f(§, u)

0 € A

and

(AL (z) f) (x, u) ^ ^ (A.,, s Lp (,, o)), f ^ + z, u) == ^ (A.,, s_ Lp (., ^ f (^ u).

5 e A o ^ A

Thus A belongs to the right ring if and only of we have for all z, x, Se <?t,

A;c, o-s Lp (z, S) == Lp ^., x} A.y+^ o

whence, putting x = 0, and writting A^ = Ao,y (yGd) we conclude, that a necessary condition is the existence of a sequence { Ay; y€^l }CR, such that

(2) Aa,, y = Ay-a; Lp (a-, y).

One sees, however, at once, that this condition is also sufficient provided, of course, { A a ; $ r r € C l } is such, that (2) defines a bounded operator on

L

(a).

Similarly one finds, that the operators in the left ring are representable as { Ky_x L p ( y , ^ ) $ x, y€<Sl) }, and conversely.

4° We recall, that the representation space H (W) of W = ind ^o consists of the collection of all complex-valued measurable functions,T^a

satisfying f(u. d) = uf {a) for all u € T and rt€^, for which

2l^

ol'

<+

provided a runs through a residue system of Cl mod T.

On any such function the action of W (d) (a€<?L) is obtained by trans- lation on the right by ft. Assume, as we can, that the total measure of Te^l, with respect to the invariant measure dr, equals 1, and let us form the central projection

p = fuR(u)d^(u).

Jrp

From what we have just said it is clear, that we have a natural identi- fication of H (W) with PL2 (<?t), such that W corresponds to the component of the right regular representation in PL2 ((ft). On the other hand, PL2 ((9l) is canonically identifiable with L2 ((Sl). Putting Pi = ^ uL«ch(u),

^T

we have, that P = {Pi S,^.; x, ?/€<Sl}. Since for any A e R , Pi A is a scalar multiple of Pi, bearing in mind what we have just seen in 3° we conclude, that the von Neumann algebra R (W) generated by W coincides in L2 (0L) with the collection of all operators having a matrix expression of the form

j ay-x P (x, y); x, ye a j (ay^C for all yea).

The commutant of R (W), corresponding to the component of the left ring of dl in PL2 ((fl)^L2 (dl) is given by the family of all bounded ope- rators, which can be written as

[ay^^(y,x); x,yea}.

Q. E. D.

REMARK 1.1. — Observe, that the reasoning employed above implies, that ind /o is the largest subrepresentation of the right regular represen-

T+a _ _

tation of d with the property, that on TC^I it restricts to a multiple of the identity map of T into itself. Analogous statement holds true upon replacing 5^0 by /o.

c. LEMMA 1.3. — With the previous notations, R (W) is a von Neumann algebra of a finite class.

Proof. — For A = { ay_sc P (^3 y); x^y^Gi} let us put / (A) = Oo. Evi- dently, f defines a linear form on R (W). To prove our lemma, it is enough to show, that Tr (AA*) == 0 implies, that A = 0, and that Tr (AA*) = Tr (A*A) for all A in R (W). One sees at once, that A* = { by_sc P (^, y) }, where bsc = a-x (^€<?l) and thus

AA* == j ^ a^ d.-y P (x, z) P (z, y) \ and A*A = j ^ d^ ay-. (3 (z, x) (3 (z, y) \

from which we infer, that

/•(AA*)=^|^p=/-(A*A)

z,

proving our lemma.

d. We recall {cf. the list of notational conventions at the end of this paper), that given a family of operators J on a Hilbert space, we shall write R (J) for the von Neumann algebra generated by the elements of J.

Let us form the subgroup <9Lo = {x\ (? [x, y)Y = 1 for all y in 0. \ of (9L. Similarly as in (a) we write <9Lo for the corresponding subgroup of el. Observe, that (flo coincides with the center Cl^ of (X.

LEMMA 1.4. — We have (R (W))^ = R (W do).

Proof. — 1° By virtue of Lemma 1.2, if A belongs to (R (W))^ = R (W) n (R (W))'

we have

A = { ay-^ (3 (x, y ) } = { by-^ (3 (y, x)

from which we conclude at once, that a^. == b^ (^€ d) and that a^ = ^{x, z^a^

for all x and z in (fl. This implies, that if a^ is nonzero, z is an element of 0io.

2° To obtain the identity claimed in our lemma, it is now sufficient to observe, that by virtue of what we saw in the proof of lemma 1.2, we have for any z€CX,

W ((z, 1)) = R (z) | H(W) = {^-.P (x, y); x, y^a {.

These two observations imply, that (R (W))^ = R (W [do). The oppo- site inclusion being trivial, our lemma is proved.

Q. E. D.

From now on we shall assume, that (9Lo is finite, in which case do is compact. We write E for the collection of all characters of (5Lo? which, when restricted to Tc^o coincide with /o [cf. (&)]. With this notation we have

COROLLARY OF LEMMA 1.4. — Writing W 0Lo = ^ •%.Py? ^d Wy for y e E

the part of W in Py H (W), W^ is a factor representation of finite class, and

w==2©w,.

% € E

Proof. — To obtain the desired conclusion, it is enough to remark, that by virtue of lemma 1.4 the center of R (W) is identical with the collection of all linear combinations of the family { P y ; ^ € E { . R (Wy) is a factor of finite class since, by Lemma 1.3, R (W) is of finite class.

Q. E. D.

e. LEMMA 1.5. — With the previous notations we have dim Py == m (^CE), where m is the (finite or infinite) index of do in d.

Proof. — We recall [cf. 4°) in (6) above], that W do is just the part of R [ do in PL2 (d). We write Ro for the regular representation of do and recall, that R do is unitarily equivalent to mRo, where m equals the (finite or infinite) index of do in d, which is the same as the index of do in d. From here to obtain, that dim Py = m (7.6=^) lt suffices to observe, that do is isomorphic to the direct product of the circle group and of a finite abelian group.

Q. E. D.

f. LEMMA 1.6. — R (Wy) is a factor of type I or II according to whether d is finite or infinite resp. (%€:E).

Proof. — Let us consider the involution S of L2 fd) == H (W) defined by (S/*) {x) =Ef{—x) [/€H(W)]. One sees at once, that if

R (W)3A = { ay-^ (3 (x, y); x, y^a j, we have

SAS={by-^(y,x);x,yea

where &, = a_, [x^ d), and thus SR (W)S == (R (W))'. If A lies, in (R (W))^

Ou 7^ 0 implies, that u belongs to do {cf. 1° in Lemma 1.4). But since x = y (do) entails P (re, y) = ^ (y, x), we can conclude, that now SAS = A*.

Therefore, in particular, S leaves invariant the subspace Py H (W) (%€SE), and denoting its part in the latter by Sy, we have

S y R ( W y ) S y = ( R ( W y ) y .

In this fashion we obtain, that R (Wy) is of type (L, In) or (Hi, Hi), according to whether m = dim Py is finite or infinite. But since m is the index of do in d, and do is assumed to be finite, we get the conclusion of our lemma.

Q. E. D.

LEMMA 1.7. — For each element o/E there is a factor representation Wy (X^E), such that Wy | (9Lo = %. I, an^

m d ^ o = = S ® W y .

T+ct ^,

r/i<0 factors R (Wy) ar^ o/ type \n if ^ is finite; otherwise they are all of type Hi.

Proof. — We have, similarly as in Lemma 1.1, ind %o = ^ ® %,

T + d o y ^ E

and thus

md_xo = ^ © ^nd_x = ^ ® Wy,

T^CZ ^ ^^ y , € E

whence the desired conclusion follows by virtue of lemma 1.6.

Q. E. D.

REMARK 1.2. — Similar result holds true if we replace %o by %o.

g. Using the previous considerations, we can complete the proof of Proposition 1.1 in the following fashion.

1° Let us consider again the character /i of Zi [cf. (a)]. The function, assigning to rceZi the complex number /i ((.r, 1)), on Zi is obviously a character of the latter; we denote it by y^. Let ^ be an arbitrary cha- racter of Z extending 7^, and let us define a. function ^ on Z by ^ (a) ==.^(^). u if a = {x, u) €Z. One verifies easily, that ^ (a) ^ (b) = ^ (a, fc) ^p (a, &), where co (a, &) = a (rr, y) if a == (n;, u), & = (y, u). We have evidently co (aao, bbo} = co (a, &) if Oo and fco are arbitrary elements in Zi.

2° We denote by Tf the group defined on the set \ (a, u); a € Z , u G T J by the law of composition

(a, u) (b, u)= (a. b, uu GO (a, &)).

The subset \ (a, 1); a € Zi ^ is a central subgroup of Z6, to be denoted again by Zi. We put M = Z^/Zi and write $ for the canonical homomorphism from Z^ onto M.

Let us define the function We on Z6 by ^Fe ((^, ^)) == ^ ( a ) . u $ ^Fe is a character of Z6. Denoting by R the right regular representation of M,

we write Ri for the part or R in the subspace; having the projection fu.R(u)d-(u)

JT

[cf. 4°) in (&)], of L^M). The representation ( R i o $ ) ( g ) ^ , of V is identically one on the subgroup { {e, u); u € T } = T^ of 7f and, by virtue of Lemma 1, [26] (p. 325), the corresponding representation on Z = V/Te is unitarily equivalent to ind /i.

Z i ^ Z

3° Let us put cX == Z / Z i ; we denote by [^. the canonical homomorphism of Z onto el. There is a function ^ on cX X cX, such that for all x, y , in Z we have ^ ([J. (rr), [J- (y)) == a (^, y). We form CX as at the start of (&) with the p just defined. Define the map ^ : Z — Cl by ^ ((rr, u)) = [J- (rr);

^ is a homomorphism, and (c/. 1°) above) oj (a, b) = ^ (C, (a), ^ (&)). Let us put for (a, u) € Z" : A ((a ?^)) = (^ (a), u) € Cl. By virtue of what we have just said, A is a homomorphism of V onto d, and its kernel coincides with Z ^ C Z6 (c/1. 2°). Let £ be the isomorphism from CX onto M = Z^/Zi such that the diagramm

is commutative. Then Ri o £ is the largest subrepresentation of the right regular representation of CX with the property, that on TC<^ it coincides with a multiple of /o [tor the latter c/*. (&)], and hence, by virtue of Remark 1.1, we have R, o £ == ind /o. Upon forming, as in {d) above,

T f - C l ^

the subgroup cXo of cX, we find, that do == ^ (Zo) = Zo/Zi, and thus do is of finite order. In this fashion, using Lemma 1.7 and Remark 1.2, with notations as loc. cit. we get, that R i o £ === V © Wy and thus also

% G E

(Rl o <I») (g) W, == ^ © (Wy o 7) 0 We.

% € E

The representation, corresponding to (Wy o X) 0 We, of Z == Z^/T,, is a factor representation of the type of Wy; when restricted to Zo === 2?, it coincides with a multiple of a character 9 in ^ [cf. (a)]. Denoting it

Ann. EC. Norm., (4), IV. — FASC. 4. 60

by U^, and by ^i the subset of 37 formed by the ^s so obtained, we get, that in the sense of unitary equivalence V = ind /i = ^ Q) U^. On

Zi^Z

1 1 ^ E ^

the other hand, we have [cf. Lemma 1.1), V == ^ © U^, and U^ is identical _ pe^i

on TcZ" to a multiple of 9. Hence we conclude finally, that ^\ -= 37

and Up == U^. By Lemma 1.7 and by what we have just said U^ is a factor representation of finite class. It is of type I if and only if eX == Z/Zi is of finite order. Since Z o / Z j is finite, this is the case if and only if Z/Zo is finite, or, since Z/Zo is isomorphic to Z/Zo, U^ is of type I if and only if the index of Z^ in Z is finite.

To complete the proof of Proposition 1.1, it is enough to observe, that y as loc. cit. is contained in 37.

Q. E. D.

REMARK 1.3. — Analogous statement holds true for a character ^ of Z ' , which on T coincides with the conjugate of the identity map of the circle group onto itself.

2. PROPOSITION 2.1. — Let Z be direct product of a sector group and of a free abelian group of finite rank, and let us consider a central extension T of Z by a one dimensional torus T. We denote by U the centralizer of the center of the connected component of F. Let ^ be a character of the center U ^ of U which, when restricted to TCU^, reduces to the identity map of the circle group onto itself. Let us put ind ^ === V (yj. With these notations,

u ^ t r

the unitary representation V (yj ofF is a factor representation of type I or II.

It is of type I if and only of the subgroup Vs To (To = connected component of the identity in F) is of a finite index in U. Finally, we have V (%J === V (%/) if and only if ^ and %/ lie on the same orbit of F in the dual of U^.

Proof. — a. We recall first (c/*. [2], p. 188), that there exist a skew symmetric bilinear form B on Z x Z with values in R, such that, putting

^ [x, y} = exp [(^/2) B (x, y)] {x, y^ Z), F is isomorphic to the group, defined on the set of all pairs (z, u) ( ^ € Z , u € T ) by the law of multiplication

(x, u) (y, v) == (x+ y , u . y . p (x, y)).

b. Given a in F, let us put ( a ' / ) (g)^^{a~^ ga) ( g G U ^ ) . We claim, that we have a y ='/ if and only of a belongs to U. To this end let us note, that

(x, u) (y, v) (x, u)-¹ == (y, ((3 (x, y)Y. u) = ( y , u) (0, (P (x, y))²)

for any pair of elements (^, u) and (y, v) in T. In particular, they commute, if and only if, (^ (^, ^/))2 == 1. Let us assume now, that a = (x, u) and that a y = = y . Then, since 1^ C U^ and y [ T == identity map of T onto itself (T = { (0, u ) } ) , we must have, in particular, (p (x, y))² = 1 whenever

(^/; ^) belongs to Fg; in other words, a must lie in U, proving our assertion.

Let us put U (y) == ind / ; we have V (7) = ind U (y), and U (/) | L^ is

u ^ t u ^ ^{l J}^^{^}

a multiple of y. We observe next, that to prove Proposition 2.1 it is enough to establish, that U (y) is a factor representation of type I or II, and that we have the first case if and only if U/U^ To is finite. In fact, since, as we saw above, U is the stable group of y in F, ind U (y) = V (7) is a

u ^ r

factor representation of the type of U (y). Thus to complete the proof of our proposition it suffices to show, that V (y) = V (y') (in the sense of unitary equivalence) if and only if y and y' differ by an action of T.

Since V (y) U^ is a multiple of the direct sum of members of the (coun- table) subset Fy of the dual of U'1, the condition is evidently necessary.

If, on the other hand, y ' : = a y ( a € T ) , then V (y') = a V (y) = V (y) [a V (y) {g) == V (y) {a~1 go)', g^T] completing the proof of our statement.

c. If W is some subset of Z, we shall write sometimes also W for the subset { ( w , l ) ; w e W } of r. On the other hand, W^- will stand for { j s ; ze Z, B (z, w) == 0 for all w in W } C Z.

Let us put Pi for the centralizer of To in T. Writting Zo for the connected component of zero in Z, and using the above notations we get easily I\ = (Zo)^.T. If Zi is the radical of the restriction of B to Z o X Z o , that is Zi = ( Z o ^ n Z o , we obtain in the same fashion U === (Zi)^.T.

Since (Zi)i1 = Zo + (Zo)i1, we have

i\.ro-((Zo)iKZo).T=u.

Clearly, Zi is the connected component of zero in (Zo)i1. Let 2 be a closed subgroup, such thet (Zo)i1 is the direct sum of Zi and of 2. Then we have also (Zi)j^ == Zo + S (direct sum). Let ^ : (Zi)]¹- -> S be the projection onto 2. Then the map W : U -> 2 defined by ^V ((^, u)) == ^ (x) is a homomorphism. Let us define the map oo : U X U -> T by

c.) (a, b) = P (<F (a), W (b)) (a, beV).

The law of composition (a, u) (&, u) = (ab, co (a, b).uu) defines a group IP on the set of all pairs { (a, u); u€T, a € U } (topologized in the obvious fashion).

Given a subgroup V of U, we shall write V'1 for its complete inverse image in LK Given a subset S c U , we shall often use the same notation for { {s, 1); 6-eS { C U6. We denote by T^ the central subgroup {{e, u);

u € T } of V6.

By virtue of the definition of co, we have

co (a. Go, b. 60) === co (a, b) (a, b e U; a,., 60 e Fo), ) [= { (a, 1); a e r o ^ C U6] is an invariant {

fies at once, that for a e F o and (&, u) € L

implying that Fo [= { (a, 1); aCFo }cV^e] is an invariant subgroup of U⁶. In fact, one verifies at once, that for a e F o and (&, u) € U6 :

(b,u)(a,l)(b,u)-^=(bab-^l).

d. Let us consider the subgroup P^ C IX Since 1^ === Z i . T, we have

r^ == T, T, == ir^ T,,

and the map, assigning to the triple (o-, a, u) (^G^. a e F i ? ^€T,,) the element cr au of 1^, is a bijection between the set lxrf,xT,, and r,.

[Observe, that here (T stands for ((cr, 1), 1) e U6' if c^eScZ, etc.] We write now %o tor the restriction of / (the latter as in the proposition) to Ff, C V\ and define a map y6' : T^ -> T by /'' (o- ai^) = /o (a). u. We claim, that yf is a character of 1^. In fact, we have

o- au. T to = v a T 6uy = O-T (r-1 a r) buu (o-, T e I-; a, b € 1^; u, v € T^), and by what we saw above, T~1 a T = a. On the other hand

^ = ((^, 1), 1) ((T, 1), 1) = ((<7 + T, P (^ T)), p~(^T))

and thus, writing a (o-, r) == ((0, ^ (o-, r)), 1) we can conclude, that

v au.r bv = v' a' i/, where

( 7 ' = o - + T € ^ , a' == a (o-, T).a.beT^ and ^ == p (o-, r).uy.

In this fashion, since /o T is the identity map, we get, that

^ (f7 GU.T ^) = %o (a (cr, T) a. b ).u.p.(5((7, T) == 7^0 (a) u.y.o (6) y = ^ (cr au).^ (r^) proving our statement.

e. Let us denote by y the Lie algebra of 1\ == Zo. T C U. Denoting by u the element of 7, such that exp (lu) =EE (0, exp (^)) ( ^ € R ) , we can identify y with Zo + R ^ , such that [z + cv, z + c'y] = B (z, z') y, and that exp {z + cu) == (s;, exp (ic))ero.

Let t) be a maximal abelian subalgebra of 7, and let us put H for the corresponding connected subgroup of Fo. If d is any element of the dual Y of Y, there is a well determined character /,/ of H satisfying jd (exp I) = exp [i (7, d)'} ( ^ € t ) ) . We recall finally, that ind jd is an

H /^ 1 o

irreducible representation of To., which we denote by V^.

Let us consider now the unitary representation W •=== ind ' /e of LK Our r^u-

next objective is to show, that if d^Y is such, that y^/] r f , = y o (==y 1^), then W is multiple of a representation p of U6, such that p | To is unitarily equivalent to V,/. To prove this, let us put K = T^ H = 2 HT,, and let us observe, that there is a character %^ of K, such that ^\T^ == j°

and y^ H == '/rf. In fact, to this end it is enough to take into conside- ration, that l ° i f ae.T\ and & e H , then ab = ba, 2° 1^ = P;nH, 3° by virtue of our choice of d, y^i TI = yf T^ We put L = K/I^ and denote by A the canonical homomorphism from K onto L. Observe, that we have

L - K/n == r? H/r? = H/r? nH - H/rf;,

and thus L is isomorphic to a vector group. The unitary representation ind 'f of K is of the form (S o A) (g)/^, where S is a continous direct sum

F f ^ K /'

of all characters of L with respect to the absolutely continous measure on L (c/*. [26], Lemma 1, p. 325). Let co be an element in L, and let us put 9 = = coo A. We claim, that there is an element meFo, such that 97^=EEm%^. In fact, since K = 2 H T^, and since elements of 2 and Fo commute with each other (both in T and IP) it is enough to find an element m in Fo, such that m ^ == y^ (y [ H). We can write the right hand side, by an appropriate choice of rf'e^'? as X^'? an(! thus n ^ffi^8

to show, that d and d' are on the same orbit of Fo in y'. But since y [ r;; FEE 1, we have ja \ T^ == ^ | 1^ and thus d ^ = d! -{\ Since [y^ y] == R u, we have the desired conclusion. Let us put p = ind %^.

K.^ U6

By what we have just seen, ind ?.y^, is unitarily equivalent to p, and thus,

R4.U6 '

by virtue of what we said above about S, W = ind y6 •==• ind (S o A) (g)^

v r^u" ' K^V-

is unitarily equivalent to a multiple of p. Therefore, to complete our proof of the above assertion, it suffices to show, that p To is unitarily equivalent to V,/ = ind y^. To this end we shall use the following propo-

ne Po

sition, which is a trivial consequence of Theorem 12.1 in [25] (p. 127).

Assume, that G is a separable locally compact group, Gi and G^ closed

subgroups of G, such that G = G i . G ^ . Let y be a character of Gi. Then, putting ^ = % | Gi n Ga, ^ We

(ind ^ [ Ga = ind y'.

VGi^G ^ G,UG^G

Taking U^ for G, K for Gi, ^ for ^ and F, for G, we get

Gi.G, - K.ro = in ro.T, = iro.T, = IP = G,

G, n G2 == H and ^ Gi n G2 =^ %,/.

Therefore

0 1^ ^=: (. ^md ^) 1 ^ -.^ ^ = ^vrf

which is the desired conclusion.

f. Let us put M = U^/Fo, and let us denote by <E> the canonical homo- morphism from U" onto M. Recalling [cf. (c)J, that IP = UT^ == SFoT,, it is easy to see, that M is identifiable to the set { (<7, u); CT e 2, u € T^

with the multiplication (a, u) (r, v) = (o + T, My ? (CT, r)), such that

< & ( ( a : , u ) , y ) ) = (y(,r),p). We show now, that $ (M") ^(U'y.r,. To this end let us observe first, that putting 2i = { cr; cy€2 and f8 (cr, r))2 = 1 for all T in 2 }, we have

Mⁱ' = = { ( ( ^ , u ) ; < 7 € 2 „ U € T } .

In this fashion, to arrive at the desired conclusion it is enough to establish, that (z, u) in U belongs to U^ if and only if z€2i + Z,. Let us write

2 = a + z" (^SS, Z o G Z o ) . Then for T € S and z'eZo we have

(p^T+z'))2^^^,^)^^^,^)^

and, evidently, the right hand side is identically one for all T and z' if and only if cre^i, and Z o S Z i .

We observe, that M is a central extension of a free abelian group of finite rank by a one dimensional torus. Let (o be a character of M^

such that M ((0, u)) == u. By virtue of Proposition 1.1 (c/. also Remark 1.3)' ind co is a factor representation of a finite class, and it is of type I if and

M l . T M .

only if M/M" is finite. By what we saw above, we have M/M^ = U'^U1')" r,.

Therefore we can conclude, that

m^ (" ° «&) = / ind u\ o $ (= B, say)

W.IUU. ^ A M J • •/

is a factor representation of finite class, and it is, since U ^ U ' / . r o is isomorphic to U/U^.Fo, of type I if and only if U^Fo has a finite index in U.

g. Let us consider now the representation B (^) W of U6 [W === ind y^;

L r^u6

cf. (e) above!. The yon Neumann algebra R (B (^) W) it generates is a multiple of R (B 0 p) == R (B) (g) tl, where 0 is the full ring of the repre- sentation space H (p) of p. Therefore, R (B (^) W) is a factor of type I or II, and we have the first case if and only if R (B) is of type I, or if and only if U / U ^ F o is finite.

By virtue of what we saw in (&), we shall have completed the proof of Proposition 2.1 at once we can show, that R (B (^) W) is unitarily equivalent to R (U (7,)), for an appropriate choice of co in the character group of M^ [such that, as before, co ((0, u)) = u; cf, (f) above]. To this end we shall use the following assertion, which is a trivial consequence of theorem 12.2 in [25] (p. 128). Assume, that G is a separable locally compact groupy Gi and G^ closed subgroups of G, such that G==Gi.G^. Let YI and ^ be characters of Gi and Ga resp.y and let us write

X'-CxilGinG.Xx.lGmGO.

Then we have

ind ^i (g) ind %a = ind ^'.

Gi -^ G Ga ^ G Gi n Ga ^ G

Let us choose for G the group U6, and for Gi and Ga (U^)6 To and 1^ resp.

We have

Gi == 1, To T, and G, = ^ T, and thus

Gi.Ga =iroT,=IP =G.

Also,

Gi n G, = 1, l1 T, == U^ T. = W.

Let G- be an element of 2,. Then (o-, u) ( u € T ) can be viewed as an element of U ^ = 2 i r g but also as an element of U2 == ^I^. Let us define the map o» : M^ -> T by <o ((cr, u)) = ^ ((cr, 1)). u; OD is a character of M^

such that (o ((0, u)) ===u ( u € T ) . In fact, if reSi, and creT, we have

(a, u) (T, u ) = (cr + T, (3 (cr, r). uy),

and thus

CO ((CT, U) (T, P)) = ^ ((CT + T, 1)) P (Or. T).l2 U.

On the other hand, in U^ :

( ^ 1 ) ( T , 1 ) = = ( ^ + T , P ( ^ T ) ) ,

and hence, since ^ T == identity map,

X ((^ 1)). X ((-, 1)) = X ((^ + ^ 1)) P (^ ^) proving our statement.

Let us suppose, that /i == <o o $ (co being chosen as above). Then y' T,, is identically one, y' F^ ^ ^e 1^ = yo = 7 1^? y' 2 = co o $ Si.

and thus, by virtue of the definition of co, is the same as y lifted from U^ to (U^y. But then, by virtue of what we said above we can conclude, that B 0 W is unitarily equivalent to U (/) lifted to IP, and thus the rings R (B (^) W) and R (U (y)) are spatially isomorphic.

Q. E. D.

REMARK 2.1. — Analogous conclusion holds true for a character y of U" which, on TcU", reduces to the conjugate of the identity map of the circle group onto itself.

REMARK 2.2. — Let us observe, that the previous reasonings imply, that U^ = r'5, an that U/Fo U^ is isomorphic to I\/r^ (for this cf, also Lemma 6.5 below). Hence V(y) is of type I if and only if the index of the center of the centralizer of the connected component of the identity of r (== r\) in r\ is finite.

3. In this section G will denote a connected and simply connected Lie group, and L a closed, connected, type I invariant subgroup of G, such that G/L is abelian. Let us recall, that by virtue of a recent result of J. Dixmier, a choice, of the indicated sort, of L is always possible {cf. [14]).

In the following, given two unitary representations pi and p^, we shall often write pi r^j p^ to express, that they are unitarily equivalent, but not necessarily identical as concrete representations. Given a set S of equivalence classes of unitary representations, we shall denote by Sc the set of the corresponding concrete representations. For a summary of the results concerning projective extensions etc. used in the sequel, the reader is referred to Section 4 (p. 18) in [2].

Let T. be a fixed element in L; we shall denote by the same letter a fixed concrete representation of the class TI. Let G^ be the stable group

of T i € L in G, and let us denote by T/' a projective extension of r. to G::A

such that

7r'- (a) ^ (b) == a (a, 6) T^' (ab) (a, b e G^) and a (a/i, &/,) == a (a, &) (Zi, Z, e L).

By virtue of our assumptions bearing on G and L, G;:/L, being a closed subgroup of the vector group G/L, is isomorphic to R ^ X Z ^ ; thus the extension cocycle a is cohomologous to a skew symmetric bilinear form, with values in T (== circle group, lifted from G^/L to G^ {cf. [2], p. 188)).

We denote by G^ the group defined on the set G ^ X T by the law of multiplication

(a, u) (6, v) == (ab, a (a, b). uu) (a, b e Gr.; ", v e T).

We assume, as we can, that a is continuous and take G^ with the product topology on G;:XT. One verifies at once, thet the subset { (7, 1); i l € S L } is a closed invariant subgroup of G^; we denote it again by L. Let us

A

put M^ = G^/L; M^ is called the Mackey group belonging to T I € L . By virtue of what proceeds, M^ satisfies the exact sequence

l ^ T ^ M ^ R - x Z ^ l .

LEMMA 3. 1. — Let A be a closed subgroup of G, such that A±!L. If p 1.9 a unitary representation of A, such that p [ L ^ TC, ^ 1uwe G^A.

Proof. — Given a € G , let us put a p {x) = p (a^ xa) ( a € A ) . Then w^ have for any a in A : a r . ^ ^ a p L ^ p | L^^T., proving our statement.

Q. E. D.

We write 0 for the canonical homomorphism from G^ onto M^ == G^/L.

Given a subset S of Gr., we shall put S" == { {s, u); ^eS, u € T }. A being assumed as above, we have

LEMMA 3.2. — There is pe(A^ ,<?u^ that p L^T:, i/an^ only if $ (A'') i^ abelian in M^.

Proof. — a. We show first the necessity. We can assume, that p | L == Ti, Then there is a continuous map f: A -> T such that T^ (a) =f{a). p (a) from where a (a, &) = /*(a) .f{b)lf{ab) ( a , & e A ) . We have furthermore /*(al) ==f{a) ( a € A , ;eL) implying, since G/L is abelian, f (ab) = f (ba) and a (a, &) = a (&, a) (a, 6 e A ) . But then

(a, u) (b, u) = (ab, a (a, b).uu) == (ba, a (b, a).uu) (Z, 1) = (b, u) (a, 12) (Z, 1) (a, &eA, Z = a-¹ 6-¹ afteL),

proving, that $ (A^) is abelian.

Ann. 'EC. Norm., (4), IV. — FASC. 4. 61

b. Let us put ^ = G^/L, and let us denote by W the canonical homomor- phism Gr. -> ^. We denote by ? the cocycle on ^ x ^ which, when lifted to G^, coincides with a. Then M^ can be realized as the group defined on the set of pairs { (c, u); c € ^ , u<ET} by the law of multi- plication

(c, u) (d, v) == (cd, [3 (c, d). uv) (c, d e 9},

and we have

<P ((a, u)) = ((V (a), u) (aeG^).

c. To show the sufficiency of our condition, let us now assume, that

$(A6) is abelian. Then, writing T^ for { ( e , u); u e T ^ c ^ A6) , there is a closed subgroup B of M^, such that $ (A6) = B X T^. Let T be the projection of $ (A6) onto B and let us put

T ((b, 1)) = (b, g (b)) (6e^(A)).

Then, writing h for 1/g we obtain

(3 (c, d) = h (c). A (a)/A (cd) (c, d e V (A))

implying, that a (a, &) = f(a). /•(&)// (a&), where f{a)=h{W (a)) (o, & € A ) . Putting, finally, p (a) = TI'- (a)//•(a) ( a € A ) , p is a representation, restric- ting on L to Tt, of A.

Q. E. D.

Given a subgroup U of G;, such that U contains T == {(e, u) }cG^

we shall write U/T for the canonical image of U in G^. If A is some subgroup of M^, we denote by A-^ its centralizer in M^.

LEMMA 3.3. — Let A and p be as in the previous lemma, and Gr, the stabi- lizer of the image of p in A. We have Gp == <& (($ (A.^^IT.

Proof. — a. Let a be some element of G^, and let us assume, that $ ((a, 1)) commutes with $ (A6). We show first, that this assumption implies, that a belongs to G^. We denote by B the smallest closed subgroup, containing a and A, of G^. Evidently, $ (B6) is abelian and hence, by virtue of Lemma 3.2, there is ^(a), with a- L = TI. Also, we can find a character y of A, such that p ~ y . ( a A) (y ( L == 1), and therefore

a p ~ cp.a(<r|A) = c».(aa | A) == <p.(o-|A)~p, implying, that a belongs to Gp.

b. We assume next, that a p < ^ p ; by virtue of Lemma 3.1 this implies aeG^. We shall show, that <t> ((a, 1)) commutes with <& (A^). We suppose again, as in (a) of Lemma 3.2, that p -| L == ^ and Ti6 A ^ / ' p (cf. foe. c^.). Then we can conclude, that a ir⁶1 A ^^ ^ A. Given any fixed a in G^, an easy computation, the details of which we leave to the reader, shows, that

(a 0 (V) = ri (b) (^ (a))-¹ ^ (^ n⁶ (a) where

Y] (b) FEE a (a, a-1). a (a-1, ^). a (a-1 b, a) (b e G^).

By virtue of what we saw above we infer from this, that with a satisfying a p ^ p w e g e t Y ] ( & ) ^ l for all & in A, or a (a, a~1) = a (a~1, &)a(a~1 b, a) ( & e A ) But this implies at once that (a, 1)~1 (&, i) (a, 1) == (a"1 ba, 1), and thus the left hand side is of the form (;, !)(&, 1) (I = a~1 b~1 ab, & € A ) , from where the conclusion is clear.

Q. E. D.

LEMMA 3.4. — Let us denote by 51 the family of all those closed, connected subgroups, containing L, of G, to which TT admits a trivial extension, inva- riant under (G^)o (= connected component of the identity in G). Then 51 contains a well defined maximal element.

Proof. — a. Let us start by observing, that the elements of 51 are contained in Gr, {cf. Lemma 3.1). We put F = M^, and show, that if A belongs to 51 we have ^ ( A ^ C l ^ . To this end we take into account, that obviously $ ((G^)o) == Fo and therefore, by virtue of our definition of 51 and Lemma 3.3, $ (A6) is contained in the centralizer 1^ of the connected component of the identity in F. But since A, and hence also A6, is connected we obtain, that $ ( A ^ C ^ o = T^ [cf. (c) in the proof of Proposition 2.1].

&. To complete our proof of Lemma 3.4, it will now be sufficient to establish, that the subgroup II = $ (F^/T of G^ belongs to 51. But : 1°

Evidently II is closed and connected; 2° ^ (IP) being abelian, TI extends to 11 trivially {cf. Lemma 3.2); 3°) If p is any such extension, by Lemma 3.3, since <& (II6) and <I> ((G^)o) commute, we have ( G ^ ) o C G p .

Q. E. D.

We denote, as in the previous section, the centralizer of the connected

/ A \

center Fg of F by U. If 11 and pe^II^ are as above, putting II == Go, we conclude by aid of Lemma 3.3. that IX •== ^ (U)/T.